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Theorem txindislem 21638
Description: Lemma for txindis 21639. (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 5356 . . 3 (∅ × ( I ‘𝐵)) = ∅
2 fvprc 6346 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
32xpeq1d 5295 . . 3 𝐴 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = (∅ × ( I ‘𝐵)))
4 simpr 479 . . . . . . . 8 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → 𝐵 = ∅)
54xpeq2d 5296 . . . . . . 7 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → (𝐴 × 𝐵) = (𝐴 × ∅))
6 xp0 5710 . . . . . . 7 (𝐴 × ∅) = ∅
75, 6syl6eq 2810 . . . . . 6 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)
87fveq2d 6356 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → ( I ‘(𝐴 × 𝐵)) = ( I ‘∅))
9 0ex 4942 . . . . . 6 ∅ ∈ V
10 fvi 6417 . . . . . 6 (∅ ∈ V → ( I ‘∅) = ∅)
119, 10ax-mp 5 . . . . 5 ( I ‘∅) = ∅
128, 11syl6eq 2810 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 = ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
13 dmexg 7262 . . . . . . . 8 ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
14 dmxp 5499 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
1514eleq1d 2824 . . . . . . . 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 445 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝐵 ≠ ∅) → ¬ (𝐴 × 𝐵) ∈ V)
19 fvprc 6346 . . . . 5 (¬ (𝐴 × 𝐵) ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
2018, 19syl 17 . . . 4 ((¬ 𝐴 ∈ V ∧ 𝐵 ≠ ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
2112, 20pm2.61dane 3019 . . 3 𝐴 ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
221, 3, 213eqtr4a 2820 . 2 𝐴 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
23 xp0 5710 . . 3 (( I ‘𝐴) × ∅) = ∅
24 fvprc 6346 . . . 4 𝐵 ∈ V → ( I ‘𝐵) = ∅)
2524xpeq2d 5296 . . 3 𝐵 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = (( I ‘𝐴) × ∅))
26 simpr 479 . . . . . . . 8 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → 𝐴 = ∅)
2726xpeq1d 5295 . . . . . . 7 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → (𝐴 × 𝐵) = (∅ × 𝐵))
28 0xp 5356 . . . . . . 7 (∅ × 𝐵) = ∅
2927, 28syl6eq 2810 . . . . . 6 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → (𝐴 × 𝐵) = ∅)
3029fveq2d 6356 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → ( I ‘(𝐴 × 𝐵)) = ( I ‘∅))
3130, 11syl6eq 2810 . . . 4 ((¬ 𝐵 ∈ V ∧ 𝐴 = ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
32 rnexg 7263 . . . . . . . 8 ((𝐴 × 𝐵) ∈ V → ran (𝐴 × 𝐵) ∈ V)
33 rnxp 5722 . . . . . . . . 9 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
3433eleq1d 2824 . . . . . . . 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 445 . . . . 5 ((¬ 𝐵 ∈ V ∧ 𝐴 ≠ ∅) → ¬ (𝐴 × 𝐵) ∈ V)
3837, 19syl 17 . . . 4 ((¬ 𝐵 ∈ V ∧ 𝐴 ≠ ∅) → ( I ‘(𝐴 × 𝐵)) = ∅)
3931, 38pm2.61dane 3019 . . 3 𝐵 ∈ V → ( I ‘(𝐴 × 𝐵)) = ∅)
4023, 25, 393eqtr4a 2820 . 2 𝐵 ∈ V → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
41 fvi 6417 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42 fvi 6417 . . . 4 (𝐵 ∈ V → ( I ‘𝐵) = 𝐵)
43 xpeq12 5291 . . . 4 ((( I ‘𝐴) = 𝐴 ∧ ( I ‘𝐵) = 𝐵) → (( I ‘𝐴) × ( I ‘𝐵)) = (𝐴 × 𝐵))
4441, 42, 43syl2an 495 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (( I ‘𝐴) × ( I ‘𝐵)) = (𝐴 × 𝐵))
45 xpexg 7125 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 × 𝐵) ∈ V)
46 fvi 6417 . . . 4 ((𝐴 × 𝐵) ∈ V → ( I ‘(𝐴 × 𝐵)) = (𝐴 × 𝐵))
4745, 46syl 17 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ( I ‘(𝐴 × 𝐵)) = (𝐴 × 𝐵))
4844, 47eqtr4d 2797 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)))
4922, 40, 48ecase 1020 1 (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  c0 4058   I cid 5173   × cxp 5264  dom cdm 5266  ran crn 5267  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by:  txindis  21639
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