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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fgraphxp | Structured version Visualization version GIF version |
Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
fgraphxp | ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fgraphopab 43192 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}) | |
2 | vex 3482 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
3 | vex 3482 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | op1std 8023 | . . . . . 6 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (1st ‘𝑥) = 𝑎) |
5 | 4 | fveq2d 6911 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎)) |
6 | 2, 3 | op2ndd 8024 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (2nd ‘𝑥) = 𝑏) |
7 | 5, 6 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏)) |
8 | 7 | rabxp 5737 | . . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} |
9 | df-3an 1088 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)) | |
10 | 9 | opabbii 5215 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
11 | 8, 10 | eqtri 2763 | . 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
12 | 1, 11 | eqtr4di 2793 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {crab 3433 〈cop 4637 {copab 5210 × cxp 5687 ⟶wf 6559 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
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 ax-un 7754 |
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-in 3970 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-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: hausgraph 43194 |
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