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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 5851 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → Fun {〈𝐴, 𝐶〉}) | |
2 | funsng 5851 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → Fun {〈𝐵, 𝐷〉}) | |
3 | 1, 2 | anim12i 588 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
4 | 3 | an4s 865 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
5 | 4 | 3adant3 1074 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
6 | dmsnopg 5524 | . . . . . 6 ⊢ (𝐶 ∈ 𝑋 → dom {〈𝐴, 𝐶〉} = {𝐴}) | |
7 | dmsnopg 5524 | . . . . . 6 ⊢ (𝐷 ∈ 𝑌 → dom {〈𝐵, 𝐷〉} = {𝐵}) | |
8 | 6, 7 | ineqan12d 3778 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ({𝐴} ∩ {𝐵})) |
9 | disjsn2 4193 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
10 | 8, 9 | sylan9eq 2664 | . . . 4 ⊢ (((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
11 | 10 | 3adant1 1072 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
12 | funun 5846 | . . 3 ⊢ (((Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉}) ∧ (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) | |
13 | 5, 11, 12 | syl2anc 691 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
14 | df-pr 4128 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
15 | 14 | funeqi 5824 | . 2 ⊢ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ↔ Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
16 | 13, 15 | sylibr 223 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 {cpr 4127 〈cop 4131 dom cdm 5038 Fun wfun 5798 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 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-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 |
This theorem is referenced by: funtpg 5856 funtpgOLD 5857 funpr 5858 fnprg 5861 structvtxvallem 25697 constr3pthlem2 26184 fpropnf1 40337 |
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