Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
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 39688 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}) | |
2 | vex 3495 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
3 | vex 3495 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | op1std 7688 | . . . . . 6 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (1st ‘𝑥) = 𝑎) |
5 | 4 | fveq2d 6667 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝐹‘(1st ‘𝑥)) = (𝐹‘𝑎)) |
6 | 2, 3 | op2ndd 7689 | . . . . 5 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → (2nd ‘𝑥) = 𝑏) |
7 | 5, 6 | eqeq12d 2834 | . . . 4 ⊢ (𝑥 = 〈𝑎, 𝑏〉 → ((𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝐹‘𝑎) = 𝑏)) |
8 | 7 | rabxp 5593 | . . 3 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} |
9 | df-3an 1081 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)) | |
10 | 9 | opabbii 5124 | . . 3 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ (𝐹‘𝑎) = 𝑏)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
11 | 8, 10 | eqtri 2841 | . 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)} |
12 | 1, 11 | syl6eqr 2871 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 〈cop 4563 {copab 5119 × cxp 5546 ⟶wf 6344 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: hausgraph 39690 |
Copyright terms: Public domain | W3C validator |