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Theorem isoselem 7268
Description: Lemma for isose 7270. (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 6038 . . . . . . . . 9 (𝑅 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
21biimpi 215 . . . . . . . 8 (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
32r19.21bi 3230 . . . . . . 7 ((𝑅 Se 𝐴𝑧𝐴) → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
43expcom 414 . . . . . 6 (𝑧𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
54adantl 482 . . . . 5 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
6 imaeq2 5995 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → (𝐻𝑥) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))))
76eleq1d 2821 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝐻𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
87imbi2d 340 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝜑 → (𝐻𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V)))
9 isofrlem.2 . . . . . . . . 9 (𝜑 → (𝐻𝑥) ∈ V)
108, 9vtoclg 3514 . . . . . . . 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 7265 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1513, 14sylan 580 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1615eleq1d 2821 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1712, 16sylibd 238 . . . . 5 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
185, 17syld 47 . . . 4 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1918ralrimdva 3147 . . 3 (𝜑 → (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
20 isof1o 7250 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
21 f1ofn 6768 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
22 sneq 4583 . . . . . . . . 9 (𝑦 = (𝐻𝑧) → {𝑦} = {(𝐻𝑧)})
2322imaeq2d 5999 . . . . . . . 8 (𝑦 = (𝐻𝑧) → (𝑆 “ {𝑦}) = (𝑆 “ {(𝐻𝑧)}))
2423ineq2d 4159 . . . . . . 7 (𝑦 = (𝐻𝑧) → (𝐵 ∩ (𝑆 “ {𝑦})) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
2524eleq1d 2821 . . . . . 6 (𝑦 = (𝐻𝑧) → ((𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2625ralrn 7020 . . . . 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 6774 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
29 forn 6742 . . . . . 6 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
3013, 20, 28, 294syl 19 . . . . 5 (𝜑 → ran 𝐻 = 𝐵)
3130raleqdv 3309 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3227, 31bitr3d 280 . . 3 (𝜑 → (∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3319, 32sylibd 238 . 2 (𝜑 → (𝑅 Se 𝐴 → ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
34 dfse2 6038 . 2 (𝑆 Se 𝐵 ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V)
3533, 34syl6ibr 251 1 (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  cin 3897  {csn 4573   Se wse 5573  ccnv 5619  ran crn 5621  cima 5623   Fn wfn 6474  ontowfo 6477  1-1-ontowf1o 6478  cfv 6479   Isom wiso 6480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-se 5576  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488
This theorem is referenced by:  isose  7270
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