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Mirrors > Home > MPE Home > Th. List > recseq | Structured version Visualization version GIF version |
Description: Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
recseq | ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrecseq3 8252 | . 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺)) | |
2 | df-recs 8318 | . 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
3 | df-recs 8318 | . 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
4 | 1, 2, 3 | 3eqtr4g 2798 | 1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 E cep 5537 Oncon0 6318 wrecscwrecs 8243 recscrecs 8317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 df-recs 8318 |
This theorem is referenced by: rdgeq1 8358 rdgeq2 8359 dfoi 9452 oieq1 9453 oieq2 9454 ordtypecbv 9458 dfac12r 10087 zorn2g 10444 ttukey2g 10457 csbrdgg 35846 aomclem3 41426 aomclem8 41431 |
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