Theorem isoselem 7341

Description: Lemma for isose 7343. (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 6104	. . . . . . . . 9 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
2	1	biimpi 219	. . . . . . . 8 ⊢ (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
3	2	r19.21bi 3257	. . . . . . 7 ⊢ ((𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
4	3	expcom 418	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
5	4	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
6		imaeq2 6060	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → (𝐻 “ 𝑥) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))))
7	6	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
8	7	imbi2d 343	. . . . . . . . 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 3523	. . . . . . . 8 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝜑 → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
11	10	com12 33	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
12	11	adantr 485	. . . . . 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 7338	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
15	13, 14	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
16	15	eleq1d 2848	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
17	12, 16	sylibd 242	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
18	5, 17	syld 48	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
19	18	ralrimdva 3165	. . 3 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
20		isof1o 7323	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
21		f1ofn 6823	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
22		sneq 4600	. . . . . . . . 9 ⊢ (𝑦 = (𝐻‘𝑧) → {𝑦} = {(𝐻‘𝑧)})
23	22	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑦 = (𝐻‘𝑧) → (◡𝑆 “ {𝑦}) = (◡𝑆 “ {(𝐻‘𝑧)}))
24	23	ineq2d 4174	. . . . . . 7 ⊢ (𝑦 = (𝐻‘𝑧) → (𝐵 ∩ (◡𝑆 “ {𝑦})) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
25	24	eleq1d 2848	. . . . . 6 ⊢ (𝑦 = (𝐻‘𝑧) → ((𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
26	25	ralrn 7085	. . . . 5 ⊢ (𝐻 Fn 𝐴 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
27	13, 20, 21, 26	4syl 20	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
28		f1ofo 6830	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
29		forn 6797	. . . . . 6 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
30	13, 20, 28, 29	4syl 20	. . . . 5 ⊢ (𝜑 → ran 𝐻 = 𝐵)
31	30	raleqdv 3323	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
32	27, 31	bitr3d 284	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
33	19, 32	sylibd 242	. 2 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
34		dfse2 6104	. 2 ⊢ (𝑆 Se 𝐵 ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V)
35	33, 34	imbitrrdi 255	1 ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3905  {csn 4590   Se wse 5614  ^◡ccnv 5662  ran crn 5664   “ cima 5666   Fn wfn 6533  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-se 5617  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547

This theorem is referenced by:  isose  7343