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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 7299 | . 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺)) | |
2 | df-recs 7355 | . 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
3 | df-recs 7355 | . 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
4 | 1, 2, 3 | 3eqtr4g 2669 | 1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 E cep 4947 Oncon0 5640 wrecscwrecs 7293 recscrecs 7354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-wrecs 7294 df-recs 7355 |
This theorem is referenced by: rdgeq1 7394 rdgeq2 7395 dfoi 8299 oieq1 8300 oieq2 8301 ordtypecbv 8305 dfac12r 8851 zorn2g 9208 ttukey2g 9221 csbrdgg 32351 aomclem3 36644 aomclem8 36649 |
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