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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 2676 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | fvopab3g.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 743 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓))) |
4 | fvopab3g.3 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 736 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒))) |
6 | 3, 5 | opelopabg 4918 | . 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 5931 | . . . . 5 ⊢ 𝐹 Fn 𝐶 |
10 | fnopfvb 6147 | . . . . 5 ⊢ ((𝐹 Fn 𝐶 ∧ 𝐴 ∈ 𝐶) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | |
11 | 9, 10 | mpan 702 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) |
12 | 8 | eleq2i 2680 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}) |
13 | 11, 12 | syl6bb 275 | . . 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 524 | . . 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 299 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 〈cop 4131 {copab 4642 Fn wfn 5799 ‘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-fn 5807 df-fv 5812 |
This theorem is referenced by: recmulnq 9665 |
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