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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 5413 | . . . 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 5413 | . . . 4 ⊢ (𝐺:𝐵–1-1→𝑋 → 𝐺:𝐵⟶𝑋) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝑋) |
7 | 3, 6 | casef 7077 | . 2 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋) |
8 | df-f1 5213 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝑋 ↔ (𝐹:𝐴⟶𝑋 ∧ Fun ◡𝐹)) | |
9 | 8 | simprbi 275 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝑋 → Fun ◡𝐹) |
10 | 1, 9 | syl 14 | . . 3 ⊢ (𝜑 → Fun ◡𝐹) |
11 | df-f1 5213 | . . . . 5 ⊢ (𝐺:𝐵–1-1→𝑋 ↔ (𝐺:𝐵⟶𝑋 ∧ Fun ◡𝐺)) | |
12 | 11 | simprbi 275 | . . . 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 7078 | . 2 ⊢ (𝜑 → Fun ◡case(𝐹, 𝐺)) |
16 | df-f1 5213 | . 2 ⊢ (case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋 ↔ (case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋 ∧ Fun ◡case(𝐹, 𝐺))) | |
17 | 7, 15, 16 | sylanbrc 417 | 1 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∩ cin 3126 ∅c0 3420 ◡ccnv 4619 ran crn 4621 Fun wfun 5202 ⟶wf 5204 –1-1→wf1 5205 ⊔ cdju 7026 casecdjucase 7072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1st 6131 df-2nd 6132 df-1o 6407 df-dju 7027 df-inl 7036 df-inr 7037 df-case 7073 |
This theorem is referenced by: djudom 7082 exmidsbthrlem 14253 |
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