Theorem rninxp 4990

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 3092	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴))
2		ssrnres 4989	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3		df-ima 4560	. . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
4	3	eleq2i 2207	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴))
5		vex 2692	. . . . 5 ⊢ 𝑦 ∈ V
6	5	elima 4894	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
7	4, 6	bitr3i 185	. . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
8	7	ralbii 2444	. 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 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ∩ cin 3075   ⊆ wss 3076   class class class wbr 3937   × cxp 4545  ran crn 4548   ↾ cres 4549   “ cima 4550

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by:  dminxp  4991  fncnv  5197