|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 8345 | . 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺)) | |
| 2 | df-recs 8411 | . 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
| 3 | df-recs 8411 | . 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
| 4 | 1, 2, 3 | 3eqtr4g 2802 | 1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 E cep 5583 Oncon0 6384 wrecscwrecs 8336 recscrecs 8410 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-frecs 8306 df-wrecs 8337 df-recs 8411 | 
| This theorem is referenced by: rdgeq1 8451 rdgeq2 8452 dfoi 9551 oieq1 9552 oieq2 9553 ordtypecbv 9557 dfac12r 10187 zorn2g 10543 ttukey2g 10556 csbrdgg 37330 aomclem3 43068 aomclem8 43073 | 
| Copyright terms: Public domain | W3C validator |