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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 40154 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}) | |
2 | vex 3444 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
3 | vex 3444 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | op1std 7681 | . . . . . 6 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (1st ‘𝑥) = 𝑎) |
5 | 4 | fveq2d 6649 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎)) |
6 | 2, 3 | op2ndd 7682 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (2nd ‘𝑥) = 𝑏) |
7 | 5, 6 | eqeq12d 2814 | . . . 4 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏)) |
8 | 7 | rabxp 5564 | . . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} |
9 | df-3an 1086 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)) | |
10 | 9 | opabbii 5097 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
11 | 8, 10 | eqtri 2821 | . 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
12 | 1, 11 | eqtr4di 2851 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 〈cop 4531 {copab 5092 × cxp 5517 ⟶wf 6320 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: hausgraph 40156 |
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