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Theorem txindislem 21341
Description: Lemma for txindis 21342. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindislem (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))

Proof of Theorem txindislem
StepHypRef Expression
1 0xp 5165 . . 3 (∅ × ( I ‘𝐵)) = ∅
2 fvprc 6144 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
32xpeq1d 5103 . . 3 𝐴 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = (∅ × ( I ‘𝐵)))
4 simpr 477 . . . . . . . 8 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → 𝐵 = ∅)
54xpeq2d 5104 . . . . . . 7 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → (𝐴 × 𝐵) = (𝐴 × ∅))
6 xp0 5515 . . . . . . 7 (𝐴 × ∅) = ∅
75, 6syl6eq 2676 . . . . . 6 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)
87fveq2d 6154 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → ( I ‘(𝐴 × 𝐵)) = ( I ‘∅))
9 0ex 4755 . . . . . 6 ∅ ∈ V
10 fvi 6213 . . . . . 6 (∅ ∈ V → ( I ‘∅) = ∅)
119, 10ax-mp 5 . . . . 5 ( I ‘∅) = ∅
128, 11syl6eq 2676 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
13 dmexg 7045 . . . . . . . 8 ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
14 dmxp 5308 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
1514eleq1d 2688 . . . . . . . 8 (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
1613, 15syl5ib 234 . . . . . . 7 (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
1716con3d 148 . . . . . 6 (𝐵 ≠ ∅ → (¬ 𝐴 ∈ V → ¬ (𝐴 × 𝐵) ∈ V))
1817impcom 446 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ≠ ∅) → ¬ (𝐴 × 𝐵) ∈ V)
19 fvprc 6144 . . . . 5 (¬ (𝐴 × 𝐵) ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
2018, 19syl 17 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ≠ ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
2112, 20pm2.61dane 2883 . . 3 𝐴 ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
221, 3, 213eqtr4a 2686 . 2 𝐴 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
23 xp0 5515 . . 3 (( I ‘𝐴) × ∅) = ∅
24 fvprc 6144 . . . 4 𝐵 ∈ V → ( I ‘𝐵) = ∅)
2524xpeq2d 5104 . . 3 𝐵 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = (( I ‘𝐴) × ∅))
26 simpr 477 . . . . . . . 8 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → 𝐴 = ∅)
2726xpeq1d 5103 . . . . . . 7 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → (𝐴 × 𝐵) = (∅ × 𝐵))
28 0xp 5165 . . . . . . 7 (∅ × 𝐵) = ∅
2927, 28syl6eq 2676 . . . . . 6 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → (𝐴 × 𝐵) = ∅)
3029fveq2d 6154 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → ( I ‘(𝐴 × 𝐵)) = ( I ‘∅))
3130, 11syl6eq 2676 . . . 4 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
32 rnexg 7046 . . . . . . . 8 ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
33 rnxp 5527 . . . . . . . . 9 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
3433eleq1d 2688 . . . . . . . 8 (𝐴 ≠ ∅ → (ran (𝐴 × 𝐵) ∈ V ↔ 𝐵 ∈ V))
3532, 34syl5ib 234 . . . . . . 7 (𝐴 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐵 ∈ V))
3635con3d 148 . . . . . 6 (𝐴 ≠ ∅ → (¬ 𝐵 ∈ V → ¬ (𝐴 × 𝐵) ∈ V))
3736impcom 446 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴 ≠ ∅) → ¬ (𝐴 × 𝐵) ∈ V)
3837, 19syl 17 . . . 4 ((¬ 𝐵 ∈ V ∧ 𝐴 ≠ ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
3931, 38pm2.61dane 2883 . . 3 𝐵 ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
4023, 25, 393eqtr4a 2686 . 2 𝐵 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
41 fvi 6213 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42 fvi 6213 . . . 4 (𝐵 ∈ V → ( I ‘𝐵) = 𝐵)
43 xpeq12 5099 . . . 4 ((( I ‘𝐴) = 𝐴 ∧ ( I ‘𝐵) = 𝐵) → (( I ‘𝐴) × ( I ‘𝐵)) = (𝐴 × 𝐵))
4441, 42, 43syl2an 494 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (( I ‘𝐴) × ( I ‘𝐵)) = (𝐴 × 𝐵))
45 xpexg 6914 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V)
46 fvi 6213 . . . 4 ((𝐴 × 𝐵) ∈ V → ( I ‘(𝐴 × 𝐵)) = (𝐴 × 𝐵))
4745, 46syl 17 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ( I ‘(𝐴 × 𝐵)) = (𝐴 × 𝐵))
4844, 47eqtr4d 2663 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
4922, 40, 48ecase 982 1 (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1992  wne 2796  Vcvv 3191  c0 3896   I cid 4989   × cxp 5077  dom cdm 5079  ran crn 5080  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858
This theorem is referenced by:  txindis  21342
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