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| Mirrors > Home > MPE Home > Th. List > rdgeq2 | 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 |
|---|---|
| rdgeq2 | ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifeq1 4529 | . . . 4 ⊢ (𝐴 = 𝐵 → if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) | |
| 2 | 1 | mpteq2dv 5244 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
| 3 | recseq 8414 | . . 3 ⊢ ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))) |
| 5 | df-rdg 8450 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
| 6 | df-rdg 8450 | . 2 ⊢ rec(𝐹, 𝐵) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
| 7 | 4, 5, 6 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∅c0 4333 ifcif 4525 ∪ cuni 4907 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Lim wlim 6385 ‘cfv 6561 recscrecs 8410 reccrdg 8449 |
| 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-mpt 5226 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 df-rdg 8450 |
| This theorem is referenced by: rdgeq12 8453 rdg0g 8467 oav 8549 itunifval 10456 hsmex 10472 ltweuz 14002 seqeq1 14045 dfrdg2 35796 finxpeq2 37388 finxpreclem6 37397 finxpsuclem 37398 |
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