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Theorem isoselem 6545
Description: Lemma for isose 6547. (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 5458 . . . . . . . . 9 (𝑅 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
21biimpi 206 . . . . . . . 8 (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
32r19.21bi 2927 . . . . . . 7 ((𝑅 Se 𝐴𝑧𝐴) → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
43expcom 451 . . . . . 6 (𝑧𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
54adantl 482 . . . . 5 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
6 imaeq2 5421 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → (𝐻𝑥) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))))
76eleq1d 2683 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝐻𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
87imbi2d 330 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝜑 → (𝐻𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V)))
9 isofrlem.2 . . . . . . . . 9 (𝜑 → (𝐻𝑥) ∈ V)
108, 9vtoclg 3252 . . . . . . . 8 ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
1110com12 32 . . . . . . 7 (𝜑 → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
1211adantr 481 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
13 isofrlem.1 . . . . . . . 8 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
14 isoini 6542 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1513, 14sylan 488 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1615eleq1d 2683 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1712, 16sylibd 229 . . . . 5 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
185, 17syld 47 . . . 4 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1918ralrimdva 2963 . . 3 (𝜑 → (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
20 isof1o 6527 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
21 f1ofn 6095 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
22 sneq 4158 . . . . . . . . 9 (𝑦 = (𝐻𝑧) → {𝑦} = {(𝐻𝑧)})
2322imaeq2d 5425 . . . . . . . 8 (𝑦 = (𝐻𝑧) → (𝑆 “ {𝑦}) = (𝑆 “ {(𝐻𝑧)}))
2423ineq2d 3792 . . . . . . 7 (𝑦 = (𝐻𝑧) → (𝐵 ∩ (𝑆 “ {𝑦})) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
2524eleq1d 2683 . . . . . 6 (𝑦 = (𝐻𝑧) → ((𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2625ralrn 6318 . . . . 5 (𝐻 Fn 𝐴 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2713, 20, 21, 264syl 19 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
28 f1ofo 6101 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
29 forn 6075 . . . . . 6 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
3013, 20, 28, 294syl 19 . . . . 5 (𝜑 → ran 𝐻 = 𝐵)
3130raleqdv 3133 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3227, 31bitr3d 270 . . 3 (𝜑 → (∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3319, 32sylibd 229 . 2 (𝜑 → (𝑅 Se 𝐴 → ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
34 dfse2 5458 . 2 (𝑆 Se 𝐵 ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V)
3533, 34syl6ibr 242 1 (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  {csn 4148   Se wse 5031  ccnv 5073  ran crn 5075  cima 5077   Fn wfn 5842  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-se 5034  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856
This theorem is referenced by:  isose  6547
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