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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 5440 | . . . 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 5440 | . . . 4 ⊢ (𝐺:𝐵–1-1→𝑋 → 𝐺:𝐵⟶𝑋) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝑋) |
7 | 3, 6 | casef 7117 | . 2 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋) |
8 | df-f1 5240 | . . . . 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 5240 | . . . . 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 7118 | . 2 ⊢ (𝜑 → Fun ◡case(𝐹, 𝐺)) |
16 | df-f1 5240 | . 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 1364 ∩ cin 3143 ∅c0 3437 ◡ccnv 4643 ran crn 4645 Fun wfun 5229 ⟶wf 5231 –1-1→wf1 5232 ⊔ cdju 7066 casecdjucase 7112 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1st 6165 df-2nd 6166 df-1o 6441 df-dju 7067 df-inl 7076 df-inr 7077 df-case 7113 |
This theorem is referenced by: djudom 7122 exmidsbthrlem 15229 |
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