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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 5328 | . . . 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 5328 | . . . 4 ⊢ (𝐺:𝐵–1-1→𝑋 → 𝐺:𝐵⟶𝑋) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝑋) |
7 | 3, 6 | casef 6973 | . 2 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋) |
8 | df-f1 5128 | . . . . 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 5128 | . . . . 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 6974 | . 2 ⊢ (𝜑 → Fun ◡case(𝐹, 𝐺)) |
16 | df-f1 5128 | . 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 3070 ∅c0 3363 ◡ccnv 4538 ran crn 4540 Fun wfun 5117 ⟶wf 5119 –1-1→wf1 5120 ⊔ cdju 6922 casecdjucase 6968 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 |
This theorem is referenced by: djudom 6978 exmidsbthrlem 13220 |
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