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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 3364 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
2 | fvopab6.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | fvopab6.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2781 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
5 | 2, 4 | anbi12d 616 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶))) |
6 | iba 517 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶))) | |
7 | 6 | bicomd 213 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓)) |
8 | moeq 3534 | . . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
9 | 8 | moani 2674 | . . . . 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 3354 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
13 | 12 | biantrur 520 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))) |
14 | 13 | opabbii 4851 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
15 | 11, 14 | eqtri 2793 | . . . 4 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
16 | 5, 7, 10, 15 | fvopab3ig 6420 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
17 | 1, 16 | sylan 569 | . 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 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∃*wmo 2619 Vcvv 3351 {copab 4846 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 |
This theorem is referenced by: (None) |
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