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Theorem isoselem 5687
 Description: Lemma for isose 5688. (Contributed by Mario Carneiro, 23-Jun-2015.)
Hypotheses
Ref Expression
isofrlem.1 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
isofrlem.2 (𝜑 → (𝐻𝑥) ∈ V)
Assertion
Ref Expression
isoselem (𝜑 → (𝑅 Se 𝐴𝑆 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.
StepHypRef Expression
1 dfse2 4880 . . . . . . . . 9 (𝑅 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
21biimpi 119 . . . . . . . 8 (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
32r19.21bi 2495 . . . . . . 7 ((𝑅 Se 𝐴𝑧𝐴) → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
43expcom 115 . . . . . 6 (𝑧𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
54adantl 273 . . . . 5 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
6 imaeq2 4845 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → (𝐻𝑥) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))))
76eleq1d 2184 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝐻𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
87imbi2d 229 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝜑 → (𝐻𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V)))
9 isofrlem.2 . . . . . . . . 9 (𝜑 → (𝐻𝑥) ∈ V)
108, 9vtoclg 2718 . . . . . . . 8 ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
1110com12 30 . . . . . . 7 (𝜑 → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
1211adantr 272 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
13 isofrlem.1 . . . . . . . 8 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
14 isoini 5685 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1513, 14sylan 279 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1615eleq1d 2184 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1712, 16sylibd 148 . . . . 5 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
185, 17syld 45 . . . 4 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1918ralrimdva 2487 . . 3 (𝜑 → (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
20 isof1o 5674 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
21 f1ofn 5334 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
22 sneq 3506 . . . . . . . . 9 (𝑦 = (𝐻𝑧) → {𝑦} = {(𝐻𝑧)})
2322imaeq2d 4849 . . . . . . . 8 (𝑦 = (𝐻𝑧) → (𝑆 “ {𝑦}) = (𝑆 “ {(𝐻𝑧)}))
2423ineq2d 3245 . . . . . . 7 (𝑦 = (𝐻𝑧) → (𝐵 ∩ (𝑆 “ {𝑦})) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
2524eleq1d 2184 . . . . . 6 (𝑦 = (𝐻𝑧) → ((𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2625ralrn 5524 . . . . 5 (𝐻 Fn 𝐴 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2713, 20, 21, 264syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
28 f1ofo 5340 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
29 forn 5316 . . . . . 6 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
3013, 20, 28, 294syl 18 . . . . 5 (𝜑 → ran 𝐻 = 𝐵)
3130raleqdv 2607 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3227, 31bitr3d 189 . . 3 (𝜑 → (∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3319, 32sylibd 148 . 2 (𝜑 → (𝑅 Se 𝐴 → ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
34 dfse2 4880 . 2 (𝑆 Se 𝐵 ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V)
3533, 34syl6ibr 161 1 (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  ∀wral 2391  Vcvv 2658   ∩ cin 3038  {csn 3495   Se wse 4219  ◡ccnv 4506  ran crn 4508   “ cima 4510   Fn wfn 5086  –onto→wfo 5089  –1-1-onto→wf1o 5090  ‘cfv 5091   Isom wiso 5092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-se 4223  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100 This theorem is referenced by:  isose  5688
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