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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 8344 | . 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺)) | |
2 | df-recs 8410 | . 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
3 | df-recs 8410 | . 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
4 | 1, 2, 3 | 3eqtr4g 2800 | 1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 E cep 5588 Oncon0 6386 wrecscwrecs 8335 recscrecs 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fv 6571 df-ov 7434 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: rdgeq1 8450 rdgeq2 8451 dfoi 9549 oieq1 9550 oieq2 9551 ordtypecbv 9555 dfac12r 10185 zorn2g 10541 ttukey2g 10554 csbrdgg 37312 aomclem3 43045 aomclem8 43050 |
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