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Mirrors > Home > ILE Home > Th. List > recseq | 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 | fveq1 5413 | . . . . . . . 8 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑎 ↾ 𝑐)) = (𝐺‘(𝑎 ↾ 𝑐))) | |
2 | 1 | eqeq2d 2149 | . . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
3 | 2 | ralbidv 2435 | . . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)) ↔ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
4 | 3 | anbi2d 459 | . . . . 5 ⊢ (𝐹 = 𝐺 → ((𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) ↔ (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))) |
5 | 4 | rexbidv 2436 | . . . 4 ⊢ (𝐹 = 𝐺 → (∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))) ↔ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))) |
6 | 5 | abbidv 2255 | . . 3 ⊢ (𝐹 = 𝐺 → {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}) |
7 | 6 | unieqd 3742 | . 2 ⊢ (𝐹 = 𝐺 → ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))}) |
8 | df-recs 6195 | . 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))} | |
9 | df-recs 6195 | . 2 ⊢ recs(𝐺) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} | |
10 | 7, 8, 9 | 3eqtr4g 2195 | 1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 {cab 2123 ∀wral 2414 ∃wrex 2415 ∪ cuni 3731 Oncon0 4280 ↾ cres 4536 Fn wfn 5113 ‘cfv 5118 recscrecs 6194 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-recs 6195 |
This theorem is referenced by: rdgeq1 6261 rdgeq2 6262 freceq1 6282 freceq2 6283 frecsuclem 6296 |
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