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Mirrors > Home > MPE Home > Th. List > fvopab3g | Structured version Visualization version GIF version |
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fvopab3g.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
fvopab3g.3 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
fvopab3g.4 | ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑) |
fvopab3g.5 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} |
Ref | Expression |
---|---|
fvopab3g | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | fvopab3g.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓))) |
4 | fvopab3g.3 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 630 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
6 | 3, 5 | opelopabg 5548 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
7 | fvopab3g.4 | . . . . . 6 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑) | |
8 | fvopab3g.5 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} | |
9 | 7, 8 | fnopab 6707 | . . . . 5 ⊢ 𝐹 Fn 𝐶 |
10 | fnopfvb 6961 | . . . . 5 ⊢ ((𝐹 Fn 𝐶 ∧ 𝐴 ∈ 𝐶) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | |
11 | 9, 10 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) |
12 | 8 | eleq2i 2831 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}) |
13 | 11, 12 | bitrdi 287 | . . 3 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
14 | 13 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
15 | ibar 528 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) | |
16 | 15 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
17 | 6, 14, 16 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 〈cop 4637 {copab 5210 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: recmulnq 11002 |
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