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Theorem imasaddvallem 16790
Description: The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasaddf.f (𝜑𝐹:𝑉onto𝐵)
imasaddf.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasaddflem.a (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
Assertion
Ref Expression
imasaddvallem ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Distinct variable groups:   𝑞,𝑝,𝐵   𝑎,𝑏,𝑝,𝑞,𝑉   · ,𝑝,𝑞   𝑋,𝑝   𝐹,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   ,𝑎,𝑏,𝑝,𝑞   𝑌,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑎,𝑏)   · (𝑎,𝑏)   𝑋(𝑞,𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem imasaddvallem
StepHypRef Expression
1 df-ov 7148 . 2 ((𝐹𝑋) (𝐹𝑌)) = ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩)
2 imasaddf.f . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
3 imasaddf.e . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
4 imasaddflem.a . . . . . 6 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
52, 3, 4imasaddfnlem 16789 . . . . 5 (𝜑 Fn (𝐵 × 𝐵))
6 fnfun 6446 . . . . 5 ( Fn (𝐵 × 𝐵) → Fun )
75, 6syl 17 . . . 4 (𝜑 → Fun )
873ad2ant1 1125 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → Fun )
9 fveq2 6663 . . . . . . . . . . 11 (𝑝 = 𝑋 → (𝐹𝑝) = (𝐹𝑋))
109opeq1d 4801 . . . . . . . . . 10 (𝑝 = 𝑋 → ⟨(𝐹𝑝), (𝐹𝑌)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
11 fvoveq1 7168 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1210, 11opeq12d 4803 . . . . . . . . 9 (𝑝 = 𝑋 → ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
1312sneqd 4569 . . . . . . . 8 (𝑝 = 𝑋 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
1413ssiun2s 4963 . . . . . . 7 (𝑋𝑉 → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
15143ad2ant2 1126 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16 fveq2 6663 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝐹𝑞) = (𝐹𝑌))
1716opeq2d 4802 . . . . . . . . . . . 12 (𝑞 = 𝑌 → ⟨(𝐹𝑝), (𝐹𝑞)⟩ = ⟨(𝐹𝑝), (𝐹𝑌)⟩)
18 oveq2 7153 . . . . . . . . . . . . 13 (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
1918fveq2d 6667 . . . . . . . . . . . 12 (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
2017, 19opeq12d 4803 . . . . . . . . . . 11 (𝑞 = 𝑌 → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
2120sneqd 4569 . . . . . . . . . 10 (𝑞 = 𝑌 → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
2221ssiun2s 4963 . . . . . . . . 9 (𝑌𝑉 → {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2322ralrimivw 3180 . . . . . . . 8 (𝑌𝑉 → ∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
24 ss2iun 4928 . . . . . . . 8 (∀𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2523, 24syl 17 . . . . . . 7 (𝑌𝑉 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
26253ad2ant3 1127 . . . . . 6 ((𝜑𝑋𝑉𝑌𝑉) → 𝑝𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2715, 26sstrd 3974 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2843ad2ant1 1125 . . . . 5 ((𝜑𝑋𝑉𝑌𝑉) → = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
2927, 28sseqtrrd 4005 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
30 opex 5347 . . . . 5 ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
3130snss 4710 . . . 4 (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ↔ {⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ )
3229, 31sylibr 235 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → ⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ )
33 funopfv 6710 . . 3 (Fun → (⟨⟨(𝐹𝑋), (𝐹𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
348, 32, 33sylc 65 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ( ‘⟨(𝐹𝑋), (𝐹𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
351, 34syl5eq 2865 1 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933  {csn 4557  cop 4563   ciun 4910   × cxp 5546  Fun wfun 6342   Fn wfn 6343  ontowfo 6346  cfv 6348  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148
This theorem is referenced by:  imasaddval  16793  imasmulval  16796  qusaddvallem  16812
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