![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isofr2 | Structured version Visualization version GIF version |
Description: A weak form of isofr 7346 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.) |
Ref | Expression |
---|---|
isofr2 | ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
2 | imassrn 6072 | . . . 4 ⊢ (𝐻 “ 𝑥) ⊆ ran 𝐻 | |
3 | isof1o 7327 | . . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
4 | f1of 6835 | . . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵) | |
5 | frn 6727 | . . . . 5 ⊢ (𝐻:𝐴⟶𝐵 → ran 𝐻 ⊆ 𝐵) | |
6 | 3, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻 ⊆ 𝐵) |
7 | 2, 6 | sstrid 3990 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ⊆ 𝐵) |
8 | ssexg 5320 | . . 3 ⊢ (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V) | |
9 | 7, 8 | sylan 578 | . 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V) |
10 | 1, 9 | isofrlem 7344 | 1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 Fr wfr 5626 ran crn 5675 “ cima 5677 ⟶wf 6542 –1-1-onto→wf1o 6545 Isom wiso 6547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-fr 5629 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |