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Theorem isoselem 6755
Description: Lemma for isose 6757. (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 5657 . . . . . . . . 9 (𝑅 Se 𝐴 ↔ ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
21biimpi 206 . . . . . . . 8 (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
32r19.21bi 3070 . . . . . . 7 ((𝑅 Se 𝐴𝑧𝐴) → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V)
43expcom 450 . . . . . 6 (𝑧𝐴 → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
54adantl 473 . . . . 5 ((𝜑𝑧𝐴) → (𝑅 Se 𝐴 → (𝐴 ∩ (𝑅 “ {𝑧})) ∈ V))
6 imaeq2 5620 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → (𝐻𝑥) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))))
76eleq1d 2824 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝐻𝑥) ∈ V ↔ (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
87imbi2d 329 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (𝑅 “ {𝑧})) → ((𝜑 → (𝐻𝑥) ∈ V) ↔ (𝜑 → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V)))
9 isofrlem.2 . . . . . . . . 9 (𝜑 → (𝐻𝑥) ∈ V)
108, 9vtoclg 3406 . . . . . . . 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 472 . . . . . 6 ((𝜑𝑧𝐴) → ((𝐴 ∩ (𝑅 “ {𝑧})) ∈ V → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) ∈ V))
13 isofrlem.1 . . . . . . . 8 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
14 isoini 6752 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1513, 14sylan 489 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑧}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
1615eleq1d 2824 . . . . . 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 3107 . . 3 (𝜑 → (𝑅 Se 𝐴 → ∀𝑧𝐴 (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
20 isof1o 6737 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
21 f1ofn 6300 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
22 sneq 4331 . . . . . . . . 9 (𝑦 = (𝐻𝑧) → {𝑦} = {(𝐻𝑧)})
2322imaeq2d 5624 . . . . . . . 8 (𝑦 = (𝐻𝑧) → (𝑆 “ {𝑦}) = (𝑆 “ {(𝐻𝑧)}))
2423ineq2d 3957 . . . . . . 7 (𝑦 = (𝐻𝑧) → (𝐵 ∩ (𝑆 “ {𝑦})) = (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})))
2524eleq1d 2824 . . . . . 6 (𝑦 = (𝐻𝑧) → ((𝐵 ∩ (𝑆 “ {𝑦})) ∈ V ↔ (𝐵 ∩ (𝑆 “ {(𝐻𝑧)})) ∈ V))
2625ralrn 6526 . . . . 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 6306 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
29 forn 6280 . . . . . 6 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
3013, 20, 28, 294syl 19 . . . . 5 (𝜑 → ran 𝐻 = 𝐵)
3130raleqdv 3283 . . . 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 5657 . 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 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cin 3714  {csn 4321   Se wse 5223  ccnv 5265  ran crn 5267  cima 5269   Fn wfn 6044  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-se 5226  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058
This theorem is referenced by:  isose  6757
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