Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f1imaen2g | GIF version |
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6768 does not need ax-setind 4519.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
f1imaen2g | ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 527 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
2 | simplr 525 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
3 | f1f 5401 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
4 | imassrn 4962 | . . . . . . 7 ⊢ (𝐹 “ 𝐶) ⊆ ran 𝐹 | |
5 | frn 5354 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
6 | 4, 5 | sstrid 3158 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) |
7 | 3, 6 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) |
8 | 7 | ad2antrr 485 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ⊆ 𝐵) |
9 | 2, 8 | ssexd 4127 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ∈ V) |
10 | f1ores 5455 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) | |
11 | 10 | ad2ant2r 506 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
12 | f1oen2g 6729 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 “ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶)) | |
13 | 1, 9, 11, 12 | syl3anc 1233 | . 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ≈ (𝐹 “ 𝐶)) |
14 | 13 | ensymd 6757 | 1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 class class class wbr 3987 ran crn 4610 ↾ cres 4611 “ cima 4612 ⟶wf 5192 –1-1→wf1 5193 –1-1-onto→wf1o 5195 ≈ cen 6712 |
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-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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-er 6509 df-en 6715 |
This theorem is referenced by: ssenen 6825 phplem4 6829 phplem4dom 6836 phplem4on 6841 fiintim 6902 |
Copyright terms: Public domain | W3C validator |