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Mirrors > Home > ILE Home > Th. List > casef1 | GIF version |
Description: The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
casef1.f | ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑋) |
casef1.g | ⊢ (𝜑 → 𝐺:𝐵–1-1→𝑋) |
casef1.disj | ⊢ (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅) |
Ref | Expression |
---|---|
casef1 | ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | casef1.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑋) | |
2 | f1f 5323 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝑋 → 𝐹:𝐴⟶𝑋) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑋) |
4 | casef1.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1→𝑋) | |
5 | f1f 5323 | . . . 4 ⊢ (𝐺:𝐵–1-1→𝑋 → 𝐺:𝐵⟶𝑋) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝑋) |
7 | 3, 6 | casef 6966 | . 2 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋) |
8 | df-f1 5123 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝑋 ↔ (𝐹:𝐴⟶𝑋 ∧ Fun ◡𝐹)) | |
9 | 8 | simprbi 273 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝑋 → Fun ◡𝐹) |
10 | 1, 9 | syl 14 | . . 3 ⊢ (𝜑 → Fun ◡𝐹) |
11 | df-f1 5123 | . . . . 5 ⊢ (𝐺:𝐵–1-1→𝑋 ↔ (𝐺:𝐵⟶𝑋 ∧ Fun ◡𝐺)) | |
12 | 11 | simprbi 273 | . . . 4 ⊢ (𝐺:𝐵–1-1→𝑋 → Fun ◡𝐺) |
13 | 4, 12 | syl 14 | . . 3 ⊢ (𝜑 → Fun ◡𝐺) |
14 | casef1.disj | . . 3 ⊢ (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅) | |
15 | 10, 13, 14 | caseinj 6967 | . 2 ⊢ (𝜑 → Fun ◡case(𝐹, 𝐺)) |
16 | df-f1 5123 | . 2 ⊢ (case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋 ↔ (case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋 ∧ Fun ◡case(𝐹, 𝐺))) | |
17 | 7, 15, 16 | sylanbrc 413 | 1 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∩ cin 3065 ∅c0 3358 ◡ccnv 4533 ran crn 4535 Fun wfun 5112 ⟶wf 5114 –1-1→wf1 5115 ⊔ cdju 6915 casecdjucase 6961 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1st 6031 df-2nd 6032 df-1o 6306 df-dju 6916 df-inl 6925 df-inr 6926 df-case 6962 |
This theorem is referenced by: djudom 6971 exmidsbthrlem 13206 |
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