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Theorem ssrnres 4787
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Proof of Theorem ssrnres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3188 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2 rnss 4586 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
31, 2ax-mp 7 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4 rnxpss 4778 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstri 3009 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6 eqss 3015 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
75, 6mpbiran 882 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8 ssid 3019 . . . . . . . 8 𝐴𝐴
9 ssv 3020 . . . . . . . 8 𝐵 ⊆ V
10 xpss12 4467 . . . . . . . 8 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
118, 9, 10mp2an 417 . . . . . . 7 (𝐴 × 𝐵) ⊆ (𝐴 × V)
12 sslin 3193 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
1311, 12ax-mp 7 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14 df-res 4377 . . . . . 6 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
1513, 14sseqtr4i 3033 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
16 rnss 4586 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴))
1715, 16ax-mp 7 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
18 sstr 3008 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
1917, 18mpan2 416 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
20 ssel 2994 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
21 vex 2605 . . . . . . . 8 𝑦 ∈ V
2221elrn2 4598 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
2320, 22syl6ib 159 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2423ancrd 319 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵)))
2521elrn2 4598 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26 elin 3156 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27 opelxp 4394 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2827anbi2i 445 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
2921opelres 4639 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴))
3029anbi1i 446 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵))
31 anass 393 . . . . . . . . 9 (((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
3230, 31bitr2i 183 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3326, 28, 323bitri 204 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3433exbii 1537 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
35 19.41v 1824 . . . . . 6 (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3625, 34, 353bitri 204 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3724, 36syl6ibr 160 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
3837ssrdv 3006 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
3919, 38impbii 124 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
407, 39bitr2i 183 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2602  cin 2973  wss 2974  cop 3403   × cxp 4363  ran crn 4366  cres 4367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377
This theorem is referenced by:  rninxp  4788
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