Theorem disjrdx 32374

Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
disjrdx.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
disjrdx.2	⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)

Assertion

Ref	Expression
disjrdx	⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑦)

Proof of Theorem disjrdx

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjrdx.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
2		f1of 6833	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴⟶𝐶)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffvelcdmda 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶)
5		f1ofveu 7408	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
6	1, 5	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7		eqcom 2735	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
8	7	reubii 3381	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
9	6, 8	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
10		disjrdx.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)
11	10	eleq2d 2815	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵))
12	4, 9, 11	rmoxfrd 32284	. . . 4 ⊢ (𝜑 → (∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
13	12	bicomd 222	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
14	13	albidv 1916	. 2 ⊢ (𝜑 → (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
15		df-disj 5108	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16		df-disj 5108	. 2 ⊢ (Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)
17	14, 15, 16	3bitr4g 314	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∃!wreu 3370  ∃*wrmo 3371  Disj wdisj 5107  ⟶wf 6538  –1-1-onto→wf1o 6541  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-disj 5108  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  tocyccntz  32859  volmeas  33844  carsggect  33932

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