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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 3185 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
2 | fvopab6.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | fvopab6.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2620 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
5 | 2, 4 | anbi12d 743 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶))) |
6 | iba 523 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶))) | |
7 | 6 | bicomd 212 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓)) |
8 | moeq 3349 | . . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵 | |
9 | 8 | moani 2513 | . . . . 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 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
13 | 12 | biantrur 526 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))) |
14 | 13 | opabbii 4649 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
15 | 11, 14 | eqtri 2632 | . . . 4 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))} |
16 | 5, 7, 10, 15 | fvopab3ig 6188 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
17 | 1, 16 | sylan 487 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶)) |
18 | 17 | 3impia 1253 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 Vcvv 3173 {copab 4642 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: (None) |
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