Theorem isoselem 7362

Description: Lemma for isose 7364. (Contributed by Mario Carneiro, 23-Jun-2015.)

Hypotheses

Ref	Expression
isofrlem.1	⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
isofrlem.2	⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)

Assertion

Ref	Expression
isoselem	⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆

Proof of Theorem isoselem

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfse2 6117	. . . . . . . . 9 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
2	1	biimpi 216	. . . . . . . 8 ⊢ (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
3	2	r19.21bi 3250	. . . . . . 7 ⊢ ((𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
4	3	expcom 413	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
5	4	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
6		imaeq2 6073	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → (𝐻 “ 𝑥) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))))
7	6	eleq1d 2825	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
8	7	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → ((𝜑 → (𝐻 “ 𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V)))
9		isofrlem.2	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)
10	8, 9	vtoclg 3553	. . . . . . . 8 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝜑 → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
11	10	com12 32	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
13		isofrlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
14		isoini 7359	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
15	13, 14	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
16	15	eleq1d 2825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
17	12, 16	sylibd 239	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
18	5, 17	syld 47	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
19	18	ralrimdva 3153	. . 3 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
20		isof1o 7344	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
21		f1ofn 6848	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
22		sneq 4635	. . . . . . . . 9 ⊢ (𝑦 = (𝐻‘𝑧) → {𝑦} = {(𝐻‘𝑧)})
23	22	imaeq2d 6077	. . . . . . . 8 ⊢ (𝑦 = (𝐻‘𝑧) → (◡𝑆 “ {𝑦}) = (◡𝑆 “ {(𝐻‘𝑧)}))
24	23	ineq2d 4219	. . . . . . 7 ⊢ (𝑦 = (𝐻‘𝑧) → (𝐵 ∩ (◡𝑆 “ {𝑦})) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
25	24	eleq1d 2825	. . . . . 6 ⊢ (𝑦 = (𝐻‘𝑧) → ((𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
26	25	ralrn 7107	. . . . 5 ⊢ (𝐻 Fn 𝐴 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
27	13, 20, 21, 26	4syl 19	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
28		f1ofo 6854	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
29		forn 6822	. . . . . 6 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
30	13, 20, 28, 29	4syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐻 = 𝐵)
31	30	raleqdv 3325	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
32	27, 31	bitr3d 281	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
33	19, 32	sylibd 239	. 2 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
34		dfse2 6117	. 2 ⊢ (𝑆 Se 𝐵 ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V)
35	33, 34	imbitrrdi 252	1 ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ∩ cin 3949  {csn 4625   Se wse 5634  ^◡ccnv 5683  ran crn 5685   “ cima 5687   Fn wfn 6555  –onto→wfo 6558  –1-1-onto→wf1o 6559  ‘cfv 6560   Isom wiso 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-se 5637  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569

This theorem is referenced by:  isose  7364