Theorem elunirn2OLD 7263

Description: Obsolete version of elfvunirn 6929 as of 12-Jan-2025. (Contributed by Thierry Arnoux, 13-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elunirn2OLD	⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Proof of Theorem elunirn2OLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6934	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐴 ∈ dom 𝐹)
2		fveq2 6897	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	eleq2d 2815	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ (𝐹‘𝐴)))
4	3	rspcev 3609	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
5	1, 4	mpancom 687	. . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
6	5	adantl 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
7		elunirn 7261	. . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
8	7	adantr 480	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
9	6, 8	mpbird 257	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067  ∪ cuni 4908  dom cdm 5678  ran crn 5679  Fun wfun 6542  ‘cfv 6548

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 5299  ax-nul 5306  ax-pr 5429

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-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556

This theorem is referenced by: (None)

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