Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3514 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
2 | fvopab6.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | fvopab6.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2834 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
5 | 2, 4 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶))) |
6 | iba 530 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶))) | |
7 | 6 | bicomd 225 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓)) |
8 | moeq 3700 | . . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
9 | 8 | moani 2637 | . . . . 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 3499 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
13 | 12 | biantrur 533 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))) |
14 | 13 | opabbii 5135 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
15 | 11, 14 | eqtri 2846 | . . . 4 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
16 | 5, 7, 10, 15 | fvopab3ig 6766 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
17 | 1, 16 | sylan 582 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
18 | 17 | 3impia 1113 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃*wmo 2620 Vcvv 3496 {copab 5130 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |