Theorem isoselem 7321

Description: Lemma for isose 7323. (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 6086	. . . . . . . . 9 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
2	1	biimpi 218	. . . . . . . 8 ⊢ (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
3	2	r19.21bi 3253	. . . . . . 7 ⊢ ((𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
4	3	expcom 417	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
5	4	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
6		imaeq2 6042	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → (𝐻 “ 𝑥) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))))
7	6	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
8	7	imbi2d 342	. . . . . . . . 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 3521	. . . . . . . 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 484	. . . . . 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 7318	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
15	13, 14	sylan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
16	15	eleq1d 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
17	12, 16	sylibd 241	. . . . 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 3161	. . 3 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
20		isof1o 7303	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
21		f1ofn 6803	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
22		sneq 4591	. . . . . . . . 9 ⊢ (𝑦 = (𝐻‘𝑧) → {𝑦} = {(𝐻‘𝑧)})
23	22	imaeq2d 6046	. . . . . . . 8 ⊢ (𝑦 = (𝐻‘𝑧) → (◡𝑆 “ {𝑦}) = (◡𝑆 “ {(𝐻‘𝑧)}))
24	23	ineq2d 4172	. . . . . . 7 ⊢ (𝑦 = (𝐻‘𝑧) → (𝐵 ∩ (◡𝑆 “ {𝑦})) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
25	24	eleq1d 2846	. . . . . 6 ⊢ (𝑦 = (𝐻‘𝑧) → ((𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
26	25	ralrn 7065	. . . . 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 6810	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
29		forn 6777	. . . . . 6 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
30	13, 20, 28, 29	4syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐻 = 𝐵)
31	30	raleqdv 3319	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
32	27, 31	bitr3d 283	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
33	19, 32	sylibd 241	. 2 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
34		dfse2 6086	. 2 ⊢ (𝑆 Se 𝐵 ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V)
35	33, 34	imbitrrdi 254	1 ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∩ cin 3903  {csn 4581   Se wse 5596  ^◡ccnv 5644  ran crn 5646   “ cima 5648   Fn wfn 6512  –onto→wfo 6515  –1-1-onto→wf1o 6516  ‘cfv 6517   Isom wiso 6518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-se 5599  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526

This theorem is referenced by:  isose  7323