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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 36807 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}) | |
2 | vex 3176 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
3 | vex 3176 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | op1std 7069 | . . . . . 6 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (1st ‘𝑥) = 𝑎) |
5 | 4 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎)) |
6 | 2, 3 | op2ndd 7070 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (2nd ‘𝑥) = 𝑏) |
7 | 5, 6 | eqeq12d 2625 | . . . 4 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏)) |
8 | 7 | rabxp 5078 | . . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} |
9 | df-3an 1033 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)) | |
10 | 9 | opabbii 4649 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
11 | 8, 10 | eqtri 2632 | . 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
12 | 1, 11 | syl6eqr 2662 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 〈cop 4131 {copab 4642 × cxp 5036 ⟶wf 5800 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
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-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: hausgraph 36809 |
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