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Mirrors > Home > ILE Home > Th. List > f1oeq123d | GIF version |
Description: Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
f1eq123d.1 | ⊢ (φ → 𝐹 = 𝐺) |
f1eq123d.2 | ⊢ (φ → A = B) |
f1eq123d.3 | ⊢ (φ → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
f1oeq123d | ⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq123d.1 | . . 3 ⊢ (φ → 𝐹 = 𝐺) | |
2 | f1oeq1 5060 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:A–1-1-onto→𝐶)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:A–1-1-onto→𝐶)) |
4 | f1eq123d.2 | . . 3 ⊢ (φ → A = B) | |
5 | f1oeq2 5061 | . . 3 ⊢ (A = B → (𝐺:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐶)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (φ → (𝐺:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐶)) |
7 | f1eq123d.3 | . . 3 ⊢ (φ → 𝐶 = 𝐷) | |
8 | f1oeq3 5062 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐺:B–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (φ → (𝐺:B–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷)) |
10 | 3, 6, 9 | 3bitrd 203 | 1 ⊢ (φ → (𝐹:A–1-1-onto→𝐶 ↔ 𝐺:B–1-1-onto→𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 –1-1-onto→wf1o 4844 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: f1oprg 5111 |
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