Theorem rninxp 5047

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 3132	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴))
2		ssrnres 5046	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3		df-ima 4617	. . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
4	3	eleq2i 2233	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴))
5		vex 2729	. . . . 5 ⊢ 𝑦 ∈ V
6	5	elima 4951	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
7	4, 6	bitr3i 185	. . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
8	7	ralbii 2472	. 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 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   ∩ cin 3115   ⊆ wss 3116   class class class wbr 3982   × cxp 4602  ran crn 4605   ↾ cres 4606   “ cima 4607

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  dminxp  5048  fncnv  5254