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Mirrors > Home > ILE Home > Th. List > fvopab3g | 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 2227 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | fvopab3g.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 465 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓))) |
4 | fvopab3g.3 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 460 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
6 | 3, 5 | opelopabg 4240 | . 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 5306 | . . . . 5 ⊢ 𝐹 Fn 𝐶 |
10 | fnopfvb 5522 | . . . . 5 ⊢ ((𝐹 Fn 𝐶 ∧ 𝐴 ∈ 𝐶) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | |
11 | 9, 10 | mpan 421 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) |
12 | 8 | eleq2i 2231 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}) |
13 | 11, 12 | bitrdi 195 | . . 3 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
14 | 13 | adantr 274 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})) |
15 | ibar 299 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) | |
16 | 15 | adantr 274 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
17 | 6, 14, 16 | 3bitr4d 219 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∃!weu 2013 ∈ wcel 2135 〈cop 3573 {copab 4036 Fn wfn 5177 ‘cfv 5182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: recmulnqg 7323 |
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