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Theorem isoselem 5402
 Description: Lemma for isose 5403. (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 4641 . . . . . . . . 9 (𝑅 Se Az A (A ∩ (𝑅 “ {z})) V)
21biimpi 113 . . . . . . . 8 (𝑅 Se Az A (A ∩ (𝑅 “ {z})) V)
32r19.21bi 2401 . . . . . . 7 ((𝑅 Se A z A) → (A ∩ (𝑅 “ {z})) V)
43expcom 109 . . . . . 6 (z A → (𝑅 Se A → (A ∩ (𝑅 “ {z})) V))
54adantl 262 . . . . 5 ((φ z A) → (𝑅 Se A → (A ∩ (𝑅 “ {z})) V))
6 imaeq2 4607 . . . . . . . . . . 11 (x = (A ∩ (𝑅 “ {z})) → (𝐻x) = (𝐻 “ (A ∩ (𝑅 “ {z}))))
76eleq1d 2103 . . . . . . . . . 10 (x = (A ∩ (𝑅 “ {z})) → ((𝐻x) V ↔ (𝐻 “ (A ∩ (𝑅 “ {z}))) V))
87imbi2d 219 . . . . . . . . 9 (x = (A ∩ (𝑅 “ {z})) → ((φ → (𝐻x) V) ↔ (φ → (𝐻 “ (A ∩ (𝑅 “ {z}))) V)))
9 isofrlem.2 . . . . . . . . 9 (φ → (𝐻x) V)
108, 9vtoclg 2607 . . . . . . . 8 ((A ∩ (𝑅 “ {z})) V → (φ → (𝐻 “ (A ∩ (𝑅 “ {z}))) V))
1110com12 27 . . . . . . 7 (φ → ((A ∩ (𝑅 “ {z})) V → (𝐻 “ (A ∩ (𝑅 “ {z}))) V))
1211adantr 261 . . . . . 6 ((φ z A) → ((A ∩ (𝑅 “ {z})) V → (𝐻 “ (A ∩ (𝑅 “ {z}))) V))
13 isofrlem.1 . . . . . . . 8 (φ𝐻 Isom 𝑅, 𝑆 (A, B))
14 isoini 5400 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) z A) → (𝐻 “ (A ∩ (𝑅 “ {z}))) = (B ∩ (𝑆 “ {(𝐻z)})))
1513, 14sylan 267 . . . . . . 7 ((φ z A) → (𝐻 “ (A ∩ (𝑅 “ {z}))) = (B ∩ (𝑆 “ {(𝐻z)})))
1615eleq1d 2103 . . . . . 6 ((φ z A) → ((𝐻 “ (A ∩ (𝑅 “ {z}))) V ↔ (B ∩ (𝑆 “ {(𝐻z)})) V))
1712, 16sylibd 138 . . . . 5 ((φ z A) → ((A ∩ (𝑅 “ {z})) V → (B ∩ (𝑆 “ {(𝐻z)})) V))
185, 17syld 40 . . . 4 ((φ z A) → (𝑅 Se A → (B ∩ (𝑆 “ {(𝐻z)})) V))
1918ralrimdva 2393 . . 3 (φ → (𝑅 Se Az A (B ∩ (𝑆 “ {(𝐻z)})) V))
20 isof1o 5390 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
21 f1ofn 5070 . . . . 5 (𝐻:A1-1-ontoB𝐻 Fn A)
22 sneq 3378 . . . . . . . . 9 (y = (𝐻z) → {y} = {(𝐻z)})
2322imaeq2d 4611 . . . . . . . 8 (y = (𝐻z) → (𝑆 “ {y}) = (𝑆 “ {(𝐻z)}))
2423ineq2d 3132 . . . . . . 7 (y = (𝐻z) → (B ∩ (𝑆 “ {y})) = (B ∩ (𝑆 “ {(𝐻z)})))
2524eleq1d 2103 . . . . . 6 (y = (𝐻z) → ((B ∩ (𝑆 “ {y})) V ↔ (B ∩ (𝑆 “ {(𝐻z)})) V))
2625ralrn 5248 . . . . 5 (𝐻 Fn A → (y ran 𝐻(B ∩ (𝑆 “ {y})) V ↔ z A (B ∩ (𝑆 “ {(𝐻z)})) V))
2713, 20, 21, 264syl 18 . . . 4 (φ → (y ran 𝐻(B ∩ (𝑆 “ {y})) V ↔ z A (B ∩ (𝑆 “ {(𝐻z)})) V))
28 f1ofo 5076 . . . . . 6 (𝐻:A1-1-ontoB𝐻:AontoB)
29 forn 5052 . . . . . 6 (𝐻:AontoB → ran 𝐻 = B)
3013, 20, 28, 294syl 18 . . . . 5 (φ → ran 𝐻 = B)
3130raleqdv 2505 . . . 4 (φ → (y ran 𝐻(B ∩ (𝑆 “ {y})) V ↔ y B (B ∩ (𝑆 “ {y})) V))
3227, 31bitr3d 179 . . 3 (φ → (z A (B ∩ (𝑆 “ {(𝐻z)})) V ↔ y B (B ∩ (𝑆 “ {y})) V))
3319, 32sylibd 138 . 2 (φ → (𝑅 Se Ay B (B ∩ (𝑆 “ {y})) V))
34 dfse2 4641 . 2 (𝑆 Se By B (B ∩ (𝑆 “ {y})) V)
3533, 34syl6ibr 151 1 (φ → (𝑅 Se A𝑆 Se B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ∩ cin 2910  {csn 3367   Se wse 4055  ◡ccnv 4287  ran crn 4289   “ cima 4291   Fn wfn 4840  –onto→wfo 4843  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-se 4056  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854 This theorem is referenced by:  isose  5403
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