Theorem rninxp 5054

Description: Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
rninxp	⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rninxp

Step	Hyp	Ref	Expression
1		dfss3 3137	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴))
2		ssrnres 5053	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3		df-ima 4624	. . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
4	3	eleq2i 2237	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴))
5		vex 2733	. . . . 5 ⊢ 𝑦 ∈ V
6	5	elima 4958	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
7	4, 6	bitr3i 185	. . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
8	7	ralbii 2476	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
9	1, 2, 8	3bitr3i 209	1 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)

Syntax hints:   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449   ∩ cin 3120   ⊆ wss 3121   class class class wbr 3989   × cxp 4609  ran crn 4612   ↾ cres 4613   “ cima 4614

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  dminxp  5055  fncnv  5264