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| Mirrors > Home > ILE Home > Th. List > fvopab6 | GIF version | ||
| Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
| Ref | Expression |
|---|---|
| fvopab6.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} |
| fvopab6.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| fvopab6.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| fvopab6 | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 2783 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
| 2 | fvopab6.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | fvopab6.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 4 | 3 | eqeq2d 2217 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
| 5 | 2, 4 | anbi12d 473 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶))) |
| 6 | iba 300 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶))) | |
| 7 | 6 | bicomd 141 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓)) |
| 8 | moeq 2948 | . . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
| 9 | 8 | moani 2124 | . . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵) |
| 10 | 9 | a1i 9 | . . . 4 ⊢ (𝑥 ∈ V → ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵)) |
| 11 | fvopab6.1 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} | |
| 12 | vex 2775 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 13 | 12 | biantrur 303 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))) |
| 14 | 13 | opabbii 4111 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
| 15 | 11, 14 | eqtri 2226 | . . . 4 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
| 16 | 5, 7, 10, 15 | fvopab3ig 5653 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
| 17 | 1, 16 | sylan 283 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
| 18 | 17 | 3impia 1203 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∃*wmo 2055 ∈ wcel 2176 Vcvv 2772 {copab 4104 ‘cfv 5271 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-sbc 2999 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 |
| This theorem is referenced by: (None) |
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