tfr1ALT - Metamath Proof Explorer

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Theorem tfr1ALT 8415

Description: Alternate proof of tfr1 8412 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypothesis**
Ref	Expression
tfrALT.1	⊢ 𝐹 = recs(𝐺)

**Assertion**
Ref	Expression
tfr1ALT	⊢ 𝐹 Fn On

**Proof of Theorem tfr1ALT**
Step	Hyp	Ref	Expression
1		epweon 7772	. 2 ⊢ E We On
2		epse 5656	. 2 ⊢ E Se On
3		tfrALT.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4		df-recs 8386	. . . 4 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
5	3, 4	eqtri 2756	. . 3 ⊢ 𝐹 = wrecs( E , On, 𝐺)
6	5	wfr1 8350	. 2 ⊢ (( E We On ∧ E Se On) → 𝐹 Fn On)
7	1, 2, 6	mp2an 691	1 ⊢ 𝐹 Fn On

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 E cep 5576 Se wse 5626 We wwe 5627 Oncon0 6364 Fn wfn 6538 wrecscwrecs 8311 recscrecs 8385

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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-frecs 8281 df-wrecs 8312 df-recs 8386

This theorem is referenced by: (None)

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