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Mirrors > Home > ILE Home > Th. List > rdgeq1 | 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 5495 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘𝑥)) = (𝐺‘(𝑔‘𝑥))) | |
2 | 1 | iuneq2d 3898 | . . . . 5 ⊢ (𝐹 = 𝐺 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))) |
3 | 2 | uneq2d 3281 | . . . 4 ⊢ (𝐹 = 𝐺 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))) |
4 | 3 | mpteq2dv 4080 | . . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))) |
5 | recseq 6285 | . . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))) → recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))) |
7 | df-irdg 6349 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) | |
8 | df-irdg 6349 | . 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))) | |
9 | 6, 7, 8 | 3eqtr4g 2228 | 1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 Vcvv 2730 ∪ cun 3119 ∪ ciun 3873 ↦ cmpt 4050 dom cdm 4611 ‘cfv 5198 recscrecs 6283 reccrdg 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-iota 5160 df-fv 5206 df-recs 6284 df-irdg 6349 |
This theorem is referenced by: omv 6434 oeiv 6435 |
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