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Mirrors > Home > ILE Home > Th. List > fnprg | GIF version |
Description: Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
fnprg | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funprg 5248 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
2 | dmpropg 5083 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant2 1014 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) |
4 | df-fn 5201 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
5 | 1, 3, 4 | sylanbrc 415 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 {cpr 3584 〈cop 3586 dom cdm 4611 Fun wfun 5192 Fn wfn 5193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-fun 5200 df-fn 5201 |
This theorem is referenced by: fprg 5679 |
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