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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 8292 | . 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺)) | |
2 | df-recs 8358 | . 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
3 | df-recs 8358 | . 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 5575 Oncon0 6356 wrecscwrecs 8283 recscrecs 8357 |
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 3063 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-xp 5678 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-iota 6487 df-fv 6543 df-ov 7399 df-frecs 8253 df-wrecs 8284 df-recs 8358 |
This theorem is referenced by: rdgeq1 8398 rdgeq2 8399 dfoi 9493 oieq1 9494 oieq2 9495 ordtypecbv 9499 dfac12r 10128 zorn2g 10485 ttukey2g 10498 csbrdgg 36115 aomclem3 41669 aomclem8 41674 |
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