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Mirrors > Home > MPE Home > Th. List > rdgeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
rdgeq1 | ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6905 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐺‘(𝑔‘∪ dom 𝑔))) | |
2 | 1 | ifeq2d 4550 | . . . . 5 ⊢ (𝐹 = 𝐺 → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔)))) |
3 | 2 | ifeq2d 4550 | . . . 4 ⊢ (𝐹 = 𝐺 → if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))) |
4 | 3 | mpteq2dv 5249 | . . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔)))))) |
5 | recseq 8412 | . . 3 ⊢ ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))) → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))))) |
7 | df-rdg 8448 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
8 | df-rdg 8448 | . 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔)))))) | |
9 | 6, 7, 8 | 3eqtr4g 2799 | 1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Vcvv 3477 ∅c0 4338 ifcif 4530 ∪ cuni 4911 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 Lim wlim 6386 ‘cfv 6562 recscrecs 8408 reccrdg 8447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-xp 5694 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-iota 6515 df-fv 6570 df-ov 7433 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 |
This theorem is referenced by: rdgeq12 8451 rdgsucmpt2 8468 frsucmpt2 8478 seqomlem0 8487 omv 8548 oev 8550 dffi3 9468 hsmex 10469 axdc 10558 seqeq2 14042 seqval 14049 precsexlemcbv 28244 seqsval 28308 seqsfn 28329 seqsp1 28331 neibastop2 36343 rdgssun 37360 exrecfnlem 37361 dffinxpf 37367 finxpeq1 37368 |
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