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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 2820 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | fvopab3g.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓))) |
4 | fvopab3g.3 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 632 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
6 | 3, 5 | opelopabg 5390 | . 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 6469 | . . . . 5 ⊢ 𝐹 Fn 𝐶 |
10 | fnopfvb 6717 | . . . . 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 2824 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}) |
13 | 11, 12 | bitrdi 290 | . . 3 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
14 | 13 | adantr 484 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
15 | ibar 532 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) | |
16 | 15 | adantr 484 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
17 | 6, 14, 16 | 3bitr4d 314 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∃!weu 2569 〈cop 4519 {copab 5089 Fn wfn 6328 ‘cfv 6333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fn 6336 df-fv 6341 |
This theorem is referenced by: recmulnq 10457 |
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