Theorem isofr2 7300

Description: A weak form of isofr 7298 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
isofr2	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Proof of Theorem isofr2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		imassrn 6038	. . . 4 ⊢ (𝐻 “ 𝑥) ⊆ ran 𝐻
3		isof1o 7279	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 6782	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		frn 6677	. . . . 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 3947	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ⊆ 𝐵)
8		ssexg 5270	. . 3 ⊢ (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
9	7, 8	sylan 581	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
10	1, 9	isofrlem 7296	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903   Fr wfr 5582  ran crn 5633   “ cima 5635  ⟶wf 6496  –1-1-onto→wf1o 6499   Isom wiso 6501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-fr 5585  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509

This theorem is referenced by: (None)