Theorem isoselem 5815

Description: Lemma for isose 5816. (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 4997	. . . . . . . . 9 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
2	1	biimpi 120	. . . . . . . 8 ⊢ (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
3	2	r19.21bi 2565	. . . . . . 7 ⊢ ((𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V)
4	3	expcom 116	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
5	4	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V))
6		imaeq2 4962	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → (𝐻 “ 𝑥) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))))
7	6	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (◡𝑅 “ {𝑧})) → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
8	7	imbi2d 230	. . . . . . . . 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 2797	. . . . . . . 8 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝜑 → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
11	10	com12 30	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V))
12	11	adantr 276	. . . . . 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 5813	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
15	13, 14	sylan 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
16	15	eleq1d 2246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
17	12, 16	sylibd 149	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐴 ∩ (◡𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
18	5, 17	syld 45	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
19	18	ralrimdva 2557	. . 3 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
20		isof1o 5802	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
21		f1ofn 5458	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
22		sneq 3602	. . . . . . . . 9 ⊢ (𝑦 = (𝐻‘𝑧) → {𝑦} = {(𝐻‘𝑧)})
23	22	imaeq2d 4966	. . . . . . . 8 ⊢ (𝑦 = (𝐻‘𝑧) → (◡𝑆 “ {𝑦}) = (◡𝑆 “ {(𝐻‘𝑧)}))
24	23	ineq2d 3336	. . . . . . 7 ⊢ (𝑦 = (𝐻‘𝑧) → (𝐵 ∩ (◡𝑆 “ {𝑦})) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})))
25	24	eleq1d 2246	. . . . . 6 ⊢ (𝑦 = (𝐻‘𝑧) → ((𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
26	25	ralrn 5650	. . . . 5 ⊢ (𝐻 Fn 𝐴 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
27	13, 20, 21, 26	4syl 18	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V))
28		f1ofo 5464	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–onto→𝐵)
29		forn 5437	. . . . . 6 ⊢ (𝐻:𝐴–onto→𝐵 → ran 𝐻 = 𝐵)
30	13, 20, 28, 29	4syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐻 = 𝐵)
31	30	raleqdv 2678	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
32	27, 31	bitr3d 190	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑧)})) ∈ V ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
33	19, 32	sylibd 149	. 2 ⊢ (𝜑 → (𝑅 Se 𝐴 → ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V))
34		dfse2 4997	. 2 ⊢ (𝑆 Se 𝐵 ↔ ∀𝑦 ∈ 𝐵 (𝐵 ∩ (◡𝑆 “ {𝑦})) ∈ V)
35	33, 34	syl6ibr 162	1 ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ∩ cin 3128  {csn 3591   Se wse 4326  ^◡ccnv 4622  ran crn 4624   “ cima 4626   Fn wfn 5207  –onto→wfo 5210  –1-1-onto→wf1o 5211  ‘cfv 5212   Isom wiso 5213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-se 4330  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221

This theorem is referenced by:  isose  5816