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Theorem isoselem 5484
Description: Lemma for isose 5485. (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 4723 . . . . . . . . 9 (𝑅 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
21biimpi 117 . . . . . . . 8 (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
32r19.21bi 2422 . . . . . . 7 ((𝑅 Se 𝐴𝑧𝐴) → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
43expcom 113 . . . . . 6 (𝑧𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
54adantl 266 . . . . 5 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
6 imaeq2 4689 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → (𝐻𝑥) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))))
76eleq1d 2120 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝐻𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
87imbi2d 223 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝜑 → (𝐻𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V)))
9 isofrlem.2 . . . . . . . . 9 (𝜑 → (𝐻𝑥) ∈ V)
108, 9vtoclg 2628 . . . . . . . 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 265 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
13 isofrlem.1 . . . . . . . 8 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
14 isoini 5482 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1513, 14sylan 271 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1615eleq1d 2120 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1712, 16sylibd 142 . . . . 5 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
185, 17syld 44 . . . 4 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
1918ralrimdva 2414 . . 3 (𝜑 → (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
20 isof1o 5472 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
21 f1ofn 5152 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
22 sneq 3411 . . . . . . . . 9 (𝑦 = (𝐻𝑧) → {𝑦} = {(𝐻𝑧)})
2322imaeq2d 4693 . . . . . . . 8 (𝑦 = (𝐻𝑧) → (𝑆 “ {𝑦}) = (𝑆 “ {(𝐻𝑧)}))
2423ineq2d 3163 . . . . . . 7 (𝑦 = (𝐻𝑧) → (𝐵 ∩ (𝑆 “ {𝑦})) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
2524eleq1d 2120 . . . . . 6 (𝑦 = (𝐻𝑧) → ((𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2625ralrn 5330 . . . . 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 5158 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
29 forn 5134 . . . . . 6 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
3013, 20, 28, 294syl 18 . . . . 5 (𝜑 → ran 𝐻 = 𝐵)
3130raleqdv 2526 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝐻(𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3227, 31bitr3d 183 . . 3 (𝜑 → (∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
3319, 32sylibd 142 . 2 (𝜑 → (𝑅 Se 𝐴 → ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V))
34 dfse2 4723 . 2 (𝑆 Se 𝐵 ↔ ∀𝑦𝐵 (𝐵 ∩ (𝑆 “ {𝑦})) ∈ V)
3533, 34syl6ibr 155 1 (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wcel 1407  wral 2321  Vcvv 2572  cin 2941  {csn 3400   Se wse 4091  ccnv 4369  ran crn 4371  cima 4373   Fn wfn 4922  ontowfo 4925  1-1-ontowf1o 4926  cfv 4927   Isom wiso 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-rab 2330  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-se 4095  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-isom 4936
This theorem is referenced by:  isose  5485
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