Theorem rnxpm 4695

Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
rnxpm	⊢ (∃x x ∈ A → ran (A × B) = B)

Distinct variable group:   x,A

Allowed substitution hint:   B(x)

Proof of Theorem rnxpm

Step	Hyp	Ref	Expression
1		df-rn 4299	. . 3 ⊢ ran (A × B) = dom ◡(A × B)
2		cnvxp 4685	. . . 4 ⊢ ◡(A × B) = (B × A)
3	2	dmeqi 4479	. . 3 ⊢ dom ◡(A × B) = dom (B × A)
4	1, 3	eqtri 2057	. 2 ⊢ ran (A × B) = dom (B × A)
5		dmxpm 4498	. 2 ⊢ (∃x x ∈ A → dom (B × A) = B)
6	4, 5	syl5eq 2081	1 ⊢ (∃x x ∈ A → ran (A × B) = B)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  ssxpbm  4699  ssxp2  4701  xpexr2m  4705  xpima2m  4711  unixpm  4796