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Theorem dom2 6191
 Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(x) and 𝐷(y), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
Hypotheses
Ref Expression
dom2.1 (x A𝐶 B)
dom2.2 ((x A y A) → (𝐶 = 𝐷x = y))
Assertion
Ref Expression
dom2 (B 𝑉AB)
Distinct variable groups:   x,y,A   x,B,y   y,𝐶   x,𝐷
Allowed substitution hints:   𝐶(x)   𝐷(y)   𝑉(x,y)

Proof of Theorem dom2
StepHypRef Expression
1 eqid 2037 . 2 A = A
2 dom2.1 . . . 4 (x A𝐶 B)
32a1i 9 . . 3 (A = A → (x A𝐶 B))
4 dom2.2 . . . 4 ((x A y A) → (𝐶 = 𝐷x = y))
54a1i 9 . . 3 (A = A → ((x A y A) → (𝐶 = 𝐷x = y)))
63, 5dom2d 6189 . 2 (A = A → (B 𝑉AB))
71, 6ax-mp 7 1 (B 𝑉AB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   class class class wbr 3755   ≼ cdom 6156 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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 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-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-dom 6159 This theorem is referenced by: (None)
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