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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 6890 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐺‘(𝑔‘∪ dom 𝑔))) | |
2 | 1 | ifeq2d 4548 | . . . . 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 4548 | . . . 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 5250 | . . 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 8378 | . . 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 8414 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
8 | df-rdg 8414 | . 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 2796 | 1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 Vcvv 3473 ∅c0 4322 ifcif 4528 ∪ cuni 4908 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Lim wlim 6365 ‘cfv 6543 recscrecs 8374 reccrdg 8413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-iota 6495 df-fv 6551 df-ov 7415 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 |
This theorem is referenced by: rdgeq12 8417 rdgsucmpt2 8434 frsucmpt2 8444 seqomlem0 8453 omv 8516 oev 8518 dffi3 9430 hsmex 10431 axdc 10520 seqeq2 13975 seqval 13982 precsexlemcbv 27892 neibastop2 35550 rdgssun 36563 exrecfnlem 36564 dffinxpf 36570 finxpeq1 36571 |
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