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Mirrors > Home > MPE Home > Th. List > funprg | Structured version Visualization version GIF version |
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
funprg | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 6552 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → Fun {〈𝐴, 𝐶〉}) | |
2 | funsng 6552 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → Fun {〈𝐵, 𝐷〉}) | |
3 | 1, 2 | anim12i 613 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
4 | 3 | an4s 658 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
5 | 4 | 3adant3 1132 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
6 | dmsnopg 6165 | . . . . . 6 ⊢ (𝐶 ∈ 𝑋 → dom {〈𝐴, 𝐶〉} = {𝐴}) | |
7 | dmsnopg 6165 | . . . . . 6 ⊢ (𝐷 ∈ 𝑌 → dom {〈𝐵, 𝐷〉} = {𝐵}) | |
8 | 6, 7 | ineqan12d 4174 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ({𝐴} ∩ {𝐵})) |
9 | disjsn2 4673 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
10 | 8, 9 | sylan9eq 2796 | . . . 4 ⊢ (((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
11 | 10 | 3adant1 1130 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
12 | funun 6547 | . . 3 ⊢ (((Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉}) ∧ (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) | |
13 | 5, 11, 12 | syl2anc 584 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
14 | df-pr 4589 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
15 | 14 | funeqi 6522 | . 2 ⊢ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ↔ Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
16 | 13, 15 | sylibr 233 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 {csn 4586 {cpr 4588 〈cop 4592 dom cdm 5633 Fun wfun 6490 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-fun 6498 |
This theorem is referenced by: funtpg 6556 funpr 6557 fnprg 6560 fpropnf1 7214 |
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