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Theorem funimassov 5886
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
funimassov ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem funimassov
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 funimass4 5438 . 2 ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) ∈ 𝐶))
2 fveq2 5387 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 5743 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2166 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54eleq1d 2184 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
65ralxp 4650 . 2 (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) ∈ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
71, 6syl6bb 195 1 ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  wral 2391  wss 3039  cop 3498   × cxp 4505  dom cdm 4507  cima 4510  Fun wfun 5085  cfv 5091  (class class class)co 5740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-ov 5743
This theorem is referenced by: (None)
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