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Mirrors > Home > MPE Home > Th. List > fvopab6 | Structured version Visualization version 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 3510 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
2 | fvopab6.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | fvopab6.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2829 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
5 | 2, 4 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶))) |
6 | iba 528 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶))) | |
7 | 6 | bicomd 224 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓)) |
8 | moeq 3695 | . . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
9 | 8 | moani 2630 | . . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ V → ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵)) |
11 | fvopab6.1 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} | |
12 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
13 | 12 | biantrur 531 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))) |
14 | 13 | opabbii 5124 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
15 | 11, 14 | eqtri 2841 | . . . 4 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
16 | 5, 7, 10, 15 | fvopab3ig 6757 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
17 | 1, 16 | sylan 580 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
18 | 17 | 3impia 1109 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃*wmo 2613 Vcvv 3492 {copab 5119 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: (None) |
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